A small remark on the Elliott conjecture

as for any distinct integers , where is the Liouville function. (The usual formulation of the conjecture also allows one to consider more general linear forms than the shifts , but for sake of discussion let us focus on the shift case.) This conjecture remains open for , though there are now some partial results when one averages either in or in the , as discussed in this recent post.

A natural generalisation of the Chowla conjecture is the Elliott conjecture. Its original formulation was basically as follows: one had

whenever were bounded completely multiplicative functions and were distinct integers, and one of the was “non-pretentious” in the sense that

for all Dirichlet characters and real numbers . It is easy to see that some condition like (2) is necessary; for instance if and has period then can be verified to be bounded away from zero as .

as for any Dirichlet character . To support this conjecture, we proved an averaged and non-asymptotic version of this conjecture which roughly speaking showed a bound of the form

whenever was an arbitrarily slowly growing function of , was sufficiently large (depending on and the rate at which grows), and one of the obeyed the condition

for some that was sufficiently large depending on , and all Dirichlet characters of period at most . As further support of this conjecture, I recently established the bound

under the same hypotheses, where is an arbitrarily slowly growing function of .

In view of these results, it is tempting to conjecture that the condition (4) for one of the should be sufficient to obtain the bound

when is large enough depending on . This may well be the case for . However, the purpose of this blog post is to record a simple counterexample for . Let’s take for simplicity. Let be a quantity much larger than but much smaller than (e.g. ), and set

For , Taylor expansion gives

and

and hence

and hence

On the other hand one can easily verify that all of the obey (4) (the restriction there prevents from getting anywhere close to ). So it seems the correct non-asymptotic version of the Elliott conjecture is the following:

Conjecture 1 (Non-asymptotic Elliott conjecture) Let be a natural number, and let be integers. Let , let be sufficiently large depending on , and let be sufficiently large depending on . Let be bounded multiplicative functions such that for some , one has

for all Dirichlet characters of conductor at most . Then

The case of this conjecture follows from the work of Halasz; in my recent paper a logarithmically averaged version of the case of this conjecture is established. The requirement to take to be as large as does not emerge in the averaged Elliott conjecture in my previous paper with Matomaki and Radziwill; it thus seems that this averaging has concealed some of the subtler features of the Elliott conjecture. (However, this subtlety does not seem to affect the asymptotic version of the conjecture formulated in that paper, in which the hypothesis is of the form (3), and the conclusion is of the form (1).)

was small (for a slowly growing function of ) if was bounded and completely multiplicative, and one had a condition of the form

for some large . However, to obtain an analogous bound for (5) it now appears that one needs to strengthen the above condition to

in order to address the counterexample in which for some between and . This seems to suggest that proving (5) (which is closely related to the case of the Chowla conjecture) could in fact be rather difficult; the estimation of (6) relied primarily of prior work of Matomaki and Radziwill which used the hypothesis (7), but as this hypothesis is not sufficient to conclude (5), some additional input must also be used.

4 comments

It seems that the generalized non pretentious restriction in conjecture 1, was constructed to avoid the above counterexample (for which ). Is it possible to relax this restriction by adding an appropriate restriction on the product (e.g. to be ) ?

At a bare minimum, one will need to be non-pretentious. For instance suppose that is a Dirichlet character. Then the function will be periodic and will typically not have mean zero (one expects a square root cancellation, but not perfect cancellation). In this case will be a Dirichlet character which will usually not be principal. One can also multiply one or more of the by for some small and still have some non-cancellation in the sum .

In principle, once one is in the “structured” case where each of the pretends to be a modulated Dirichlet character , one can write as a Dirichlet convolution of this modulated character with another arithmetic function that pretends to be 1, and one can evaluate the sum more or less explicitly (though to deal with some error terms one may need something like a generalised Elliott-Halberstam conjecture, or else do some smoothing in the x variable). This should in principle let one refine the condition yielding cancellation in the sum to obtain a near-optimal criterion for this cancellation. (This is analogous to the situation involving finding linear patterns in dense sets, where there is a crude “Cauchy-Schwarz complexity” criterion that handles all the cases in which a certain high order Gowers norm is small, but then there is a more refined “true complexity” criterion that uses some more advanced additive combinatorics (such as the equidistribution theory of nilsequences) to find a more optimal criterion to get equidistribution.)

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